On common zeros of Legendre's associated functions
Abstract: In this paper it is proved that any two given Legendre associated functions and , where is an integer and where one of the integers m or s may be 0 (and ), have either no zero in common or exactly one common zero, namely . An auxiliary result states that the zeros of known to lie in the open interval lie in fact in the open interval , where are the two zeros of which is one of the coefficients in the Legendre associated equation satisfied by . Some monotonicity behavior of is simultaneously described.
The proof of the main result is based on properties of Prüfer polar coordinates.
-  B. C. Goodwin & N. H. J. Lacroix, "A further study of the holoblastic cleavage field," J. Theoret. Biol. (To appear.)
-  E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922
-  Hans Sagan, Boundary and eigenvalue problems in mathematical physics, John Wiley & Sons, Inc., New York-London, 1961. MR 0118932
- B. C. Goodwin & N. H. J. Lacroix, "A further study of the holoblastic cleavage field," J. Theoret. Biol. (To appear.)
- E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea, New York, 1955. MR 0064922 (16:356i)
- H. Sagan, Boundary and Eigenvalue Problems in Mathematical Physics, Wiley, New York, 1966. MR 0118932 (22:9701)
Retrieve articles in Mathematics of Computation with MSC: 33A45
Retrieve articles in all journals with MSC: 33A45
Keywords: Legendre functions, associated Legendre functions, Legendre's associated functions, zeros of orthogonal functions
Article copyright: © Copyright 1984 American Mathematical Society